If $f\in L^2(\mathbb{R}^n)$, will $ \lim_{r\to\infty} \int_{\partial B_n(0,r)} |f|^2 dS$ also be finite?

Question

If a function $f$ is in $L^2(\mathbb{R}^n)$, will $$ \lim_{r\to\infty} \int_{\partial B_n(0,r)} |f|^2 dS $$ also be finite?

If so, can this be generalized to any $L^p$ space? If not, what are some counter-examples?